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Erschienen in: Neural Computing and Applications 7/2016

01.10.2016 | Original Article

Stability in distribution of stochastic delay recurrent neural networks with Markovian switching

verfasst von: Enwen Zhu, George Yin, Quan Yuan

Erschienen in: Neural Computing and Applications | Ausgabe 7/2016

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Abstract

This paper investigates the stability in distribution of stochastic delay recurrent neural networks with Markovian switching. Using Lyapunov function and stochastic analysis techniques, sufficient conditions on the stability in distribution are given. For such recurrent neural networks, it reveals that the limit distribution of transition probability for segment process associated with solution process is indeed a unique ergodic invariant probability measure. Moreover, a numerical example is also provided to demonstrate the effectiveness and applicability of the theoretical results.

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Literatur
1.
Zurück zum Zitat Haykin S (1994) Neural networks. Prentice-Hall, Upper Saddle RiverMATH Haykin S (1994) Neural networks. Prentice-Hall, Upper Saddle RiverMATH
2.
Zurück zum Zitat Liu D, Michel A (1993) Celluar neural networks for associative memories. IEEE Trans Circuits Syst II 40(2):119–121MathSciNetCrossRef Liu D, Michel A (1993) Celluar neural networks for associative memories. IEEE Trans Circuits Syst II 40(2):119–121MathSciNetCrossRef
4.
Zurück zum Zitat Hirasawa K, Mabu S, Hu J (2006) Propagation and control of stochastic signals through universal learning networks. Neural Netw 19(4):487–499CrossRefMATH Hirasawa K, Mabu S, Hu J (2006) Propagation and control of stochastic signals through universal learning networks. Neural Netw 19(4):487–499CrossRefMATH
5.
Zurück zum Zitat Cao J, Wang J (2004) Absolute exponential stability of recurrent neural networks with Lipschitz-continuous activation functions and time delays. Neural Netw 17(3):379–390CrossRefMATH Cao J, Wang J (2004) Absolute exponential stability of recurrent neural networks with Lipschitz-continuous activation functions and time delays. Neural Netw 17(3):379–390CrossRefMATH
6.
Zurück zum Zitat Li L, Huang L (2009) Dynamical behaviors of a class of recurrent neural networks with discontinuous neuron activations. Appl Math Model 33(12):4326–4336MathSciNetCrossRefMATH Li L, Huang L (2009) Dynamical behaviors of a class of recurrent neural networks with discontinuous neuron activations. Appl Math Model 33(12):4326–4336MathSciNetCrossRefMATH
7.
Zurück zum Zitat Huang Y, Zhang H, Wang Z (2012) Dynamical stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions. Neurocomputing 91:21–28CrossRef Huang Y, Zhang H, Wang Z (2012) Dynamical stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions. Neurocomputing 91:21–28CrossRef
8.
Zurück zum Zitat Wan L, Sun J (2005) Mean square exponential stability of stochastic delayed Hopfield neural networks. Phys Lett A 343(4):306–318CrossRefMATH Wan L, Sun J (2005) Mean square exponential stability of stochastic delayed Hopfield neural networks. Phys Lett A 343(4):306–318CrossRefMATH
9.
Zurück zum Zitat Sun Y, Cao J (2007) pth moment exponential stability of stochastic recurrent neural networks with time-varying delays. Nonlinear Anal Real 8(4):1171–1185MathSciNetCrossRefMATH Sun Y, Cao J (2007) pth moment exponential stability of stochastic recurrent neural networks with time-varying delays. Nonlinear Anal Real 8(4):1171–1185MathSciNetCrossRefMATH
10.
Zurück zum Zitat Huang C, He Y, Huang L, Zhu W (2008) pth moment stability analysis of stochastic recurrent neural networks with time-varying delays. Inf Sci 178(9):2194–2203MathSciNetCrossRefMATH Huang C, He Y, Huang L, Zhu W (2008) pth moment stability analysis of stochastic recurrent neural networks with time-varying delays. Inf Sci 178(9):2194–2203MathSciNetCrossRefMATH
11.
Zurück zum Zitat Ma L, Da F (2009) Mean-square exponential stability of stochastic Hopfield neural networks with time-varying discrete and distributed delays. Phys Lett A 373(25):2154–2161MathSciNetCrossRefMATH Ma L, Da F (2009) Mean-square exponential stability of stochastic Hopfield neural networks with time-varying discrete and distributed delays. Phys Lett A 373(25):2154–2161MathSciNetCrossRefMATH
12.
Zurück zum Zitat Li B, Xu D (2009) Mean square asymptotic behavior of stochastic neural networks with infinitely distributed delays. Neurocomputing 72(13–15):3311–3317CrossRef Li B, Xu D (2009) Mean square asymptotic behavior of stochastic neural networks with infinitely distributed delays. Neurocomputing 72(13–15):3311–3317CrossRef
13.
Zurück zum Zitat Chen W, Zheng W (2010) Robust stability analysis for stochastic neural networks with time-varying delay. IEEE Trans Neural Netw 21(3):508–514CrossRef Chen W, Zheng W (2010) Robust stability analysis for stochastic neural networks with time-varying delay. IEEE Trans Neural Netw 21(3):508–514CrossRef
14.
Zurück zum Zitat Song Q (2011) Stochastic dissipativity analysis on discrete-time neural networks with time-varying. Neurocomputing 74(5):838–845CrossRef Song Q (2011) Stochastic dissipativity analysis on discrete-time neural networks with time-varying. Neurocomputing 74(5):838–845CrossRef
15.
Zurück zum Zitat Wang Z, Liu Y, Liu X (2006) Exponential stability of delayed recurrent neural networks with Markovian jumping parameters. Phys Lett A 356(4–5):346–352CrossRefMATH Wang Z, Liu Y, Liu X (2006) Exponential stability of delayed recurrent neural networks with Markovian jumping parameters. Phys Lett A 356(4–5):346–352CrossRefMATH
16.
Zurück zum Zitat Lou X, Cui B (2007) Delay-dependent stochastic stability of delayed Hopfield neural networks with Markovian jump parameters. J Math Anal Appl 328(1):316–326MathSciNetCrossRefMATH Lou X, Cui B (2007) Delay-dependent stochastic stability of delayed Hopfield neural networks with Markovian jump parameters. J Math Anal Appl 328(1):316–326MathSciNetCrossRefMATH
17.
Zurück zum Zitat Liu Y, Wang Z, Liu X (2008) On delay-dependent robust exponential stability of stochastic neural networks with mixed time delays and Markovian switching. Nonlinear Dyn 54(3):199–212MathSciNetCrossRefMATH Liu Y, Wang Z, Liu X (2008) On delay-dependent robust exponential stability of stochastic neural networks with mixed time delays and Markovian switching. Nonlinear Dyn 54(3):199–212MathSciNetCrossRefMATH
18.
Zurück zum Zitat Liu Y, Wang Z, Liang J, Liu X (2009) Stability and synchronization of discrete-time Markovian jumping neural networks with mixed mode-dependent time delays. IEEE Trans Neural Netw 20(7):1102–1116CrossRef Liu Y, Wang Z, Liang J, Liu X (2009) Stability and synchronization of discrete-time Markovian jumping neural networks with mixed mode-dependent time delays. IEEE Trans Neural Netw 20(7):1102–1116CrossRef
19.
Zurück zum Zitat Shen Y, Wang J (2009) Almost sure exponential stability of recurrent neural networks with Markov switching. IEEE Trans Neural Netw 20(5):840–855CrossRef Shen Y, Wang J (2009) Almost sure exponential stability of recurrent neural networks with Markov switching. IEEE Trans Neural Netw 20(5):840–855CrossRef
20.
Zurück zum Zitat Wu Z, Su H, Chu J (2010) State estimation for discrete Markovian jumping neural networks with delay. Neurocomputing 73(10–12):2247–2254CrossRef Wu Z, Su H, Chu J (2010) State estimation for discrete Markovian jumping neural networks with delay. Neurocomputing 73(10–12):2247–2254CrossRef
21.
Zurück zum Zitat Zhu E, Zhang H, Zou J (2010) Stability analysis of recurrent neural networks with random delay and Markovian switching. J. Inequal. Appl. 2010, Article ID 191546, 12 pages Zhu E, Zhang H, Zou J (2010) Stability analysis of recurrent neural networks with random delay and Markovian switching. J. Inequal. Appl. 2010, Article ID 191546, 12 pages
22.
Zurück zum Zitat Balasubramaniam P, Lakshmanan S (2010) State estimation for Markovian jumping recurrent neural networks with interval time-varying delays. Nonlinear Dyn 60(4):661–675MathSciNetCrossRefMATH Balasubramaniam P, Lakshmanan S (2010) State estimation for Markovian jumping recurrent neural networks with interval time-varying delays. Nonlinear Dyn 60(4):661–675MathSciNetCrossRefMATH
23.
Zurück zum Zitat Chen Y, Zheng W (2012) Stochastic state estimation for neural networks with distributed delays and Markovian jump. Neural Netw 25:14–20MathSciNetCrossRefMATH Chen Y, Zheng W (2012) Stochastic state estimation for neural networks with distributed delays and Markovian jump. Neural Netw 25:14–20MathSciNetCrossRefMATH
24.
25.
Zurück zum Zitat Tino P, Cernansky M, Benuskova L (2004) Markovian architectural bias of recurrent neural networks. IEEE Trans Neural Netw 15(1):6–15CrossRef Tino P, Cernansky M, Benuskova L (2004) Markovian architectural bias of recurrent neural networks. IEEE Trans Neural Netw 15(1):6–15CrossRef
26.
Zurück zum Zitat Mao X, Yuan C (2006) Stochastic differential equations with Markovian switching. Imperial College Press, LondonCrossRefMATH Mao X, Yuan C (2006) Stochastic differential equations with Markovian switching. Imperial College Press, LondonCrossRefMATH
27.
28.
Zurück zum Zitat Yuan C, Mao X (2003) Asymptotic stability in distribution of stochastic differential equations with Markovian switching. Stoch Process Appl 103(2):277–291MathSciNetCrossRefMATH Yuan C, Mao X (2003) Asymptotic stability in distribution of stochastic differential equations with Markovian switching. Stoch Process Appl 103(2):277–291MathSciNetCrossRefMATH
29.
Zurück zum Zitat Yuan C, Zou J, Mao X (2003) Stability in distribution of stochastic differential delay equations with Markovian switching. Syst Control Lett 50(3):195–207MathSciNetCrossRefMATH Yuan C, Zou J, Mao X (2003) Stability in distribution of stochastic differential delay equations with Markovian switching. Syst Control Lett 50(3):195–207MathSciNetCrossRefMATH
30.
Zurück zum Zitat Bao J, Hou Z, Yuan C (2009) Stability in distribution of neutral stochastic differential delay equations with Markovian switching. Stat Probabil Lett 79(15):1663–1673MathSciNetCrossRefMATH Bao J, Hou Z, Yuan C (2009) Stability in distribution of neutral stochastic differential delay equations with Markovian switching. Stat Probabil Lett 79(15):1663–1673MathSciNetCrossRefMATH
31.
Zurück zum Zitat Hu G, Wang K (2012) Stability in distribution of neutral stochastic functional differential equations with Markovian switching. J Math Anal Appl 385(2):757–769MathSciNetCrossRefMATH Hu G, Wang K (2012) Stability in distribution of neutral stochastic functional differential equations with Markovian switching. J Math Anal Appl 385(2):757–769MathSciNetCrossRefMATH
32.
Zurück zum Zitat Mao X, Matasov A, Pinunovskiy A (2000) Stochastic differential delay equations with Markovian switching. Bernoulli 6(1):73–90MathSciNetCrossRefMATH Mao X, Matasov A, Pinunovskiy A (2000) Stochastic differential delay equations with Markovian switching. Bernoulli 6(1):73–90MathSciNetCrossRefMATH
33.
Zurück zum Zitat Bao J, Yin G, Yuan C, Wang L (2014) Exponential ergodicity for retarded stochastic differential equations. Appl Anal 93(11):2330–2349MathSciNetCrossRefMATH Bao J, Yin G, Yuan C, Wang L (2014) Exponential ergodicity for retarded stochastic differential equations. Appl Anal 93(11):2330–2349MathSciNetCrossRefMATH
34.
Zurück zum Zitat Bao J, Yuan C (2014) Numerical approximation of stationary distributions for stochastic partial differential equations. J Appl Probab 51(3):858–873MathSciNetCrossRefMATH Bao J, Yuan C (2014) Numerical approximation of stationary distributions for stochastic partial differential equations. J Appl Probab 51(3):858–873MathSciNetCrossRefMATH
35.
Zurück zum Zitat Bao J, Yin G, Yuan C (2014) Ergodicity for functional stochastic differential equations and applications. Nonlinear Anal 98:66–82MathSciNetCrossRefMATH Bao J, Yin G, Yuan C (2014) Ergodicity for functional stochastic differential equations and applications. Nonlinear Anal 98:66–82MathSciNetCrossRefMATH
Metadaten
Titel
Stability in distribution of stochastic delay recurrent neural networks with Markovian switching
verfasst von
Enwen Zhu
George Yin
Quan Yuan
Publikationsdatum
01.10.2016
Verlag
Springer London
Erschienen in
Neural Computing and Applications / Ausgabe 7/2016
Print ISSN: 0941-0643
Elektronische ISSN: 1433-3058
DOI
https://doi.org/10.1007/s00521-015-2013-x

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